Graph Theory in Networks

1. Graph Theory in Networks Lecture 4, 9/9/04 EE 228A, Fall 2004 Rajarshi Gupta University of California, Berkeley

2. Plan for Graph Segment • Lecture 2 – Thu (Sep 2, 2004) • Paths and Routing • Cycles and Protection • Matching and Switching • Lecture 3 – Tue (Sep 7, 2004) • Coloring and Capacity • Trees and Broadcast, Multicast • Lecture 4 – Thu (Sep 9, 2004) • Complete example: Capacity in Ad-Hoc Networks • Lectures 8 & 9 – (Sep 23 & 28, 2004) • Student Presentations (have you signed up ?)

3. Goal Support quality of service for flows over ad-hoc networks Collaborators: • John Musacchio • Zhanfeng Jia • Prof. Jean Walrand

4. Ad-Hoc Networks • No base station • Multi-hop transmissions • Distributed and dynamic operations

5. Application Scenarios Disaster Relief Convention Center

6. Overview • Introduction and Motivation • QoS in Ad-Hoc Networks • Model and Related Work • Row Constraints • Clique Constraints • Computing Cliques • Implementation of Algorithms • Interference-based QoS Routing

7. QoS for Flows • Want to support flows with quality (bandwidth) requirements • Aspects of the problem • Maximum capacity in a network • Feasibility of a given set of flows • Available capacity once flows are assigned • Routing a given set of flows

8. Random vs Arbitrary Network • Capacity of ad-hoc networks • Random/homogenous topology, traffic matrix • Asymptotic bounds on capacity • Our Approach • Arbitrary topology, traffic matrix • Graph theoretic model • Feasibility of given set of flows • Distributed, localized and dynamic algorithm Gupta+Kumar (2000), Grossglauser+Tse (2002), El Gamal et. al. (2003)

9. What’s the problem with ad-hoc networks ? Ans: Interference • In wired networks, all links may be used simultaneously • In Ad-Hoc networks, neighboring links interfere • Interference Range (Ix) > Transmission Range (Tx)

10. Conflict Graph: L1 L1 Interference Radius L2 L2 L3 L3 Conflict Graph Three Links: F1 + F2 <= C and F2 + F3 <= C Two Links: F1 + F2 <= C Single Link: F1 <= C

11. L2 L1 L3 L5 L4 Independent Set Solution • Construct Conflict Graph • Identify All Maximal Independent Sets • {L1, L3} , {L1, L4} {L2, L4} , {L2, L5} , {L3, L5} • Write Constraints such that • Only one Independent Set “on” at a time • QoS requirements met for flow at each link “A New Model forPacketScheduling in Multihop Wireless Networks”, H. Luo, S. Lu, and V. Bhargavan, ACM Mobicom 2000.

12. Issues with Independent Sets • Shown to be necessary and sufficientfor existence of global feasible schedule • But scales poorly • Need centralized information • Finding all maximal independent sets is exponential • Takes 10’s of minutes for simple graph (<100 links) • Want distributed and sufficient constraints that can be computed quickly in a large network "Impact of Interference on Multi-hop Wireless Network Performance”, K. Jain, J. Padhye, V. N. Padmanabhan, and L. Qiu, ACM Mobicom 2003.

13. Overview • Introduction and Motivation • QoS in Ad-Hoc Networks • Model and Related Work • Row Constraints • Clique Constraints • Computing Cliques • Implementation of Algorithms • Interference-based QoS Routing

14. Conflict Graph: L1 L1 Interference Radius L2 L2 L3 L3 Conflict Graph Three Links: F1 + F2 <= C and F2 + F3 <= C Two Links: F1 + F2 <= C Single Link: F1 <= C Alternatively: F1 + F2 + F3 <= C

15. Row Constraints • At Node 2: F2 + F1 <= C • At Node 1: F1 + F2 + F3 + F4 + F5 <= C • Each row in the Conflict Graph incidence matrix yields a constraint • Proved to be sufficient for existence of feasible schedule • Often too pessimistic • F2 = F3 = F4 = F5 = C possible • Row constraints allow only F2 = F3 = F4 = F5 = C/4

16. Sufficiency of Row Constraints: Proof • Assume each weight Fi is integral (else take ) where T is number of slots • Transform CG  CGF • Replace each node i with Ki fully connected nodes • Color this graph • Each node will be scheduled for requisite number of slots • Neighboring nodes will be scheduled for disjoint slots • Need to achieve coloring in T colors/slots • Greedy algorithm • Color each node with smallest available color • Can always find such a color since sum of colors of all neighbors (row constraints) < T

17. Overview • Introduction and Motivation • QoS in Ad-Hoc Networks • Model and Related Work • Row Constraints • Clique Constraints • Computing Cliques • Implementation of Algorithms • Interference-based QoS Routing

18. Cliques • Observe • Cliques in CG are local structures (IS are global) • Only one node in a clique may be active at once • Definitions • Clique = Complete Subgraph • Maximal Clique = Clique not a subset of any other Maximal Cliques: ABC, BCEF, CDF

19. L2 L1 L3 L5 L4 Clique Constraints Clique • Identify All Maximal Cliques • {L1, L2}, {L1, L5} , {L2, L3}, {L3, L4}, {L4, L5} • Write Constraints • Only one member of a Clique can be on at once • F1+ F2 <= C, F1+ F5 <= C, ... • Necessary conditions for a feasible schedule [MSR 2003]

20. L2 L1 L3 L5 L4 Insufficiency of Clique Constraints • But, clique constraints are not sufficient • F1=F2=F3=F4=F5 = C/2 satisfy clique constraints • But, we see that only 2 of 5 nodes may be on at once • F1=F2=F3=F4=F5 = 2C/5 is the max possible allocation • Sufficient only for ‘Perfect Graphs’

21. Sufficiency using Cliques: Proof • Equivalent weighted coloring problem • Transform CG  CGF (as with Row Constraints) • Replace each node i by clique of size Fi • Color CGf with fewest colors • Observe • Schedule of a clique = color allocation for nodes in it • Capacity of a clique = total number of colors used (T) • Chromatic number Clique number • is the largest clique in CGF

22. Imperfection Ratio • Imperfection Ratio is the ratio between the weighted Chromatic and Clique numbers • Supremum over all weight (flow) vectors • Bounded when the underlying graph is UDG • Feasible schedule exists if scaled clique constraints are satisfied on a conflict graph • Scale capacity of each link by • So, “Graph Imperfection I”, S. Gerke and C. McDiarmid, Journal of Combinatorial Theory, Series B, vol. 83 (2001), pp. 58-78.

23. Extensions to Realistic Networks • Earlier results valid for CG that are unit disk graph • Variance in interference range • Model interference range varying between [x,1] • Then, need to scale the clique constraints by • Obstructions in network • Consider virtual CGV without obstructions • Feasible schedule in CGV implies schedule in CG • Satisfy scaled clique constraints in CGV

24. Constraint-based Algorithms • Background Computation • Local link state exchange (position, flows) • Distributedly compute maximal cliques in CG • Constraint-based approach • Check sufficiency with row constraints • Estimate capacity using scaled clique constraints • Useful for • Admission Control • Clustering • Routing

25. Overview • Introduction and Motivation • QoS in Ad-Hoc Networks • Model and Related Work • Row Constraints • Clique Constraints • Computing Cliques [time ?] • Implementation of Algorithms • Interference-based QoS Routing

26. Approximate the interference of a link by a circle centred at mid-point Since Ix > Tx, the extra area is small Representing a Link by its Center

27. Computing Cliques • General algorithms are centralized and exponential • Propose computationally simple heuristic approximation (for ad-hoc networks) • Key observations for an interference CG • All links sharing cliques with this link must lie within a circle of radius Ix (interference range) • All links that lie within a circle of diameter Ix must form a clique Harary+Ross (1957), Bierstone (1960s), Augustson et. al. (1970), Bron+Kerbosch (1973)

28. Heuristic Clique Algorithm • Use a disk of radius Ix/2 to scan a disk of radius Ix around link • Each position of scanning disk generates a clique • Heuristically shrink set of cliques • Only remember previous clique • Check containment • Can further shrink to set of maximal cliques • Brute force check against all existing cliques

29. Overview • Introduction and Motivation • QoS in Ad-Hoc Networks • Model and Related Work • Row Constraints • Clique Constraints • Computing Cliques • Implementation of Algorithms • Interference-based QoS Routing

30. 0 kbps 500 kbps 1000 kbps 0 33 18 27 4 10 14 3 11 46 17 48 100% 0.5 43 15 20 37 38 23 6 1 39 41 47 21 40 50% 5 22 44 36 16 29 9 49 1.5 28 7 1 26 12 42 13 34 0% 45 2 35 50 25 31 8 2 24 19 30 32 2.5 0 0.5 1 1.5 2 2.5 Choose Destination Routing… Click on bar to choose flow rate Choose Source Y position in km X position in km

31. 0 kbps 500 kbps 1000 kbps Choose Next Source Choose Destination Click on bar to choose flow rate Routing…

33. 0 kbps 500 kbps 1000 kbps Choose Next Source Choose Destination Click on bar to choose flow rate Flow Rejected. Insufficient Resources

34. Overview • Introduction and Motivation • QoS in Ad-Hoc Networks • Model and Related Work • Row Constraints • Clique Constraints • Computing Cliques • Implementation of Algorithms • Simulations of 802.11b • Interference-based QoS Routing

35. Shortest Path Methods ?? • 1-3 is widest path from node 1 to 3 • Consider path from 1 to 5 • Path 1-3-4-5: FA+FD+FE<=C, so f<=C/3 • Path 1-2-3-4-5: FB+FC<=C, FC+FD<=C, FD+FE<=C, so f<=C/2 • Violates Bellman’s principle of optimality • Does not conform to distributed algorithm extending path hop by hop • Distributed algorithm unlikely to be optimal • Work with distributed heuristic algorithms

36. Ad-Hoc Shortest Widest Path • Recall Lec 2: distributed SWP is sub-optimal • Solution • At each node, remember every possible combination of path length and width • Exponential algoritm :-( • Approximation • Remember a few sets of optimal paths • ASWP (remembers only best set) • 2-ASWP (remembers two) • -ASWP (optimal solution)

37. SWP Tradeoffs • Width vs Resource utilization • Denote width of a path as the max flow possible on that path • When introducing a new flow, clearly width -ASWP  4-ASWP  2-ASWP  ASWP  SP • But consider resources utilized by path. Then, -ASWP  4-ASWP  2-ASWP  ASWP  SP • -ASWP may not be best in the long run

38. SWP Tradeoffs (contd) • Short Paths • Take least resources • Tend to crowd middle of network • Wide Paths • Use up too much resources • Computation intensive • Turns out (simulations) that ASWP is typically good enough to provide long term benefits

39. Source Routing Heuristic • Link state exchange allows src to know • Topology • Available capacity on all links i • New flow (src, dest, bw) arrives • Choose several candidate paths by source routing • Shortest Path (SP) • SP compliment • Shortest Feasible Path • Shortest Widest Path (SWP) • Weighted Path Cost (OSPF) • Send probe packets along each path • Final path chosen and confirmed by destination

40. 4 6 3 5 7 2 8 1 Distributed Path Evaluation • Paths compared via suitable (monotone) metrics • Probe packets • Evaluate clique constraints along path • Check for violated constraints • Accumulate path metric • Destination chooses amongst multiple viable paths • Once path confirmed, avlbw updated in network F23+F34+F45 <= avlbw F45+F56+F67 <= avlbw

41. 4 6 3 5 7 2 8 1 Row Constraints • Keep everything same • Except evaluate row constraints along path • Guaranteed to find distributed schedule • Could be employed only for high priority flows F23 + F34 + F45+ F56 + F67 <= avlbw

42. Measurement-based • Link state protocol used to compute cliques as before • But measurement-based avlbw instead of clique-based • avlbw = Idle / (Transmitting + Listening + Noisy + Idle) • Accounts for • distributed scheduling • invisible interference • Cliques still used by probe packets to estimate effect of new flow on avlbw • e.g. new flow uses 3 links on my worst clique, so need 3 x flowbw • Once flow admitted, true effect recallibrated by avlbw measurements

43. Lessons from this lecture • Important to model critical phenomenon as appropriate graph (CG) • Map physical behavior to graph feature • Utilize graph theory and results – Cliques, IS • Opens up many other related avenues, e.g. routing (ASWP)